Transcript 10.3 Inscribed Angles

Inscribed Angles
Challenge Problem
Given: 𝐺𝐻 and the tangent l
intersecting at 𝐻 on circle ⨀𝐸
Prove: 𝑚∠𝐺𝐻𝐼 =
1
2
𝑚𝐺𝐹𝐻
F
G
l
I
E
H
D
F
G
l
I
1. 𝐺𝐻 and tangent line l intersect at 𝐻 on ⨀ 𝐸 (given)
2. Construct diameter 𝐻𝐷 intersecting ⨀𝐸 at 𝐷.
3. ∠𝐷𝐻𝐼 is a right angle (tangent and radius are perpendicular)
4. 𝐷𝐺𝐻 is a semicircle of measure 180 (def of semicircle)
5. 𝑚∠𝐷𝐻𝐺 +𝑚∠𝐺𝐻𝐼 = 𝑚∠𝐷𝐻𝐼 (angle addition)
6. 𝑚𝐷𝐺 + 𝑚𝐺𝐹𝐻 = 𝑚𝐷𝐺𝐻 (arc addition)
7. 90 = 𝑚∠𝐷𝐻𝐺 + 𝑚∠𝐺𝐻𝐼 (substitution)
8. 180 = 𝑚𝐷𝐺 + 𝑚𝐺𝐹𝐻 (substitution)
1
9. 90 = 2 (𝑚𝐷𝐺 + 𝑚𝐺𝐹𝐻) (division)
10. 𝑚∠𝐷𝐻𝐺 + 𝑚∠𝐺𝐻𝐼 =
1
2
𝑚𝐷𝐺 +
1
𝑚𝐺𝐹𝐻
2
(substitution and
distribution)
11. 𝑚∠𝐷𝐻𝐺 =
12. 𝑚∠𝐺𝐻𝐼 =
1
𝑚𝐷𝐺 (Inscribed angle
2
1
𝑚𝐺𝐹𝐻 (subtraction)
2
theorem)
E
H
Objectives
• Use inscribed angles to solve problems.
• Use properties of inscribed polygons.
Using Inscribed Angles
• An inscribed angle is an
angle whose vertex is on
a circle and whose sides
contain chords of the
circle. The arc that lies in
the interior of an inscribed
angle and has endpoints
on the angle is called the
intercepted arc of the
angle.
intercepted arc
inscribed angle
Measure of an Inscribed Angle
A
• If an angle is
inscribed in a
circle, then its
measure is one
half the measure
of its intercepted
arc.
𝑚𝐴𝐷𝐵 = ½𝑚 AB

C
D
B
Finding Measures of Arcs and Inscribed Angles
• Find the measure
of the blue arc or
angle.
S
R

m Q TS = 2mQRS =
2(90°) = 180°
T
Q
Finding Measures of Arcs and Inscribed Angles
W
• Find the measure
of the blue arc or
angle.

Z
Y
X
m ZWX = 2mZYX = 2(115°) = 230°
Ex. 1: Finding Measures of Arcs and Inscribed Angles
• Find the measure
of the blue arc or
angle.


N
100°
M
m NMP = ½ m NP
P
½ (100°) = 50°
Ex. 2: Comparing Measures of Inscribed Angles
A
• Find mACB,
mADB, and
mAEB.
E
The measure of each
angle is half the
measure of AB
m AB = 60°, so the
measure of each
angle is 30°

B

D
C
Theorem
A
• If two inscribed angles
of a circle intercept
the same arc, then the
angles are congruent.
• C  D
D
B
C
Ex. 3: Finding the Measure of an Angle
G
• It is given that
mE = 75°. What
is mF?

• E and F both
intercept GH , so
E  F. So,
mF = mE = 75°
E
75°
F
H
Using Properties of Inscribed Polygons
• If all of the vertices of
a polygon lie on a
circle, the polygon is
inscribed in the circle
and the circle is
circumscribed about
the polygon. The
polygon is an
inscribed polygon
and the circle is a
circumscribed circle.
Theorem
• If a right triangle is inscribed in a circle,
then the hypotenuse is a diameter of
the circle. Conversely, if one side of
an inscribed triangle is a diameter of
the circle, then the triangle is a right
triangle and the angle opposite the
B
diameter is the right angle.
• B is a right angle if and only if AC is
a diameter of the circle.
A
C
Theorem
• A quadrilateral can be
inscribed in a circle if and
E
only if its opposite angles are
supplementary.
• D, E, F, and G lie on some
circle, C, if and only if
𝑚𝐷 + 𝑚𝐹 = 180°
D
• 𝑎𝑛𝑑 𝑚𝐸 + 𝑚𝐺 = 180°
F
C
G
Ex. 5: Using Theorems
• Find the value of
each variable.
• AB is a diameter.
So, C is a right
angle and mC =
90°
• 2𝑥° = 90°
• 𝑥 = 45
B
Q
A
2x°
C
Ex. 5: Using Theorems
• Find the value of each
variable.
• DEFG is inscribed in a
circle, so opposite
angles are
supplementary.
• 𝑚𝐷 + 𝑚𝐹 = 180°
• 𝑧 + 80 = 180
• 𝑧 = 100
D
E
z°
120°
80°
y°
G
F
Ex. 5: Using Theorems
• Find the value of each
variable.
• DEFG is inscribed in a
circle, so opposite
angles are
supplementary.
• 𝑚𝐸 + 𝑚𝐺 = 180°
• 𝑦 + 120 = 180
• 𝑦 = 60
D
E
z°
120°
80°
y°
G
F
Ex. 6: Using an Inscribed Quadrilateral
• In the diagram, ABCD is
inscribed in circle P.
Find the measure of
each angle.
A
2y°
D
3y°
3x° B
• ABCD is inscribed in a
circle, so opposite
2x°
angles are
C
supplementary.
To solve this system of linear equations,
• 3x + 3y = 180
you can solve the first equation for y to
• 5x + 2y = 180
get y = 60 – x. Substitute this
expression into the second equation.
Ex. 6: Using an Inscribed Quadrilateral
•
•
•
•
•
•
5x + 2y = 180.
5x + 2 (60 – x) = 180
5x + 120 – 2x = 180
3x = 60
x = 20
y = 60 – 20 = 40
Write the second equation.
Substitute 60 – x for y.
Distributive Property.
Subtract 120 from both sides.
Divide each side by 3.
Substitute and solve for y.
x = 20 and y = 40, so mA = 80°, mB = 60°,
mC = 100°, and mD = 120°
Angle Measures and Segment Lengths
in Circles
Objectives:
1. To find the measures of s formed by chords,
secants, & tangents.
2. To find the lengths of segments associated with
circles.
Secants
F
B
Secant – A line that
intersects a circle in exactly
2 points.
•𝐸𝐹 or 𝐴𝐵 are secants
•𝐴𝐵 is a chord
A
E
Theorem. The measure of an  formed by 2 lines that
intersect inside a circle is
1
𝑚1 = (𝑥 + 𝑦)
2
𝒚°
Measure of intercepted arcs
𝟏
𝒙°
Theorem. The measure of an  formed by 2 lines
that intersect outside a circle is
Smaller Arc
𝟏
𝒎𝟏 = (𝒙 − 𝒚)
𝟐
3 cases:
Larger Arc
2 Secants:
Tangent & a Secant
1
1
1
y°
y°
y°
x°
2 Tangents
x°
x°
Ex.1 & 2:
Find the measure of arc 𝑥.
Find the 𝑚𝑥.
𝒙°
𝒙°
92°
104°
68°
94°
112°
1
𝑚1 = (𝑥 + 𝑦)
2
1
94 = (112 + 𝑥)
2
188 = (112 + 𝑥)
76° = 𝑥
268°
𝑚𝑥 = ½(𝑥 − 𝑦)
𝑚𝑥 = ½(268 − 92)
𝑚𝑥 = ½(176)
𝑚𝑥 = 88°
Lengths of Secants, Tangents, & Chords
2 Chords
2 Secants
Tangent & Secant
𝑦
𝑎
𝑐
𝑡
𝑏
𝑥
𝑑
𝑎•𝑏 = 𝑐•𝑑
𝑧
𝑧
𝑤
𝑦
𝑤(𝑤 + 𝑥) = 𝑦(𝑦 + 𝑧)
𝑡2 = 𝑦(𝑦 + 𝑧)
Ex. 3 & 4
Find length of 𝑥.
Find the length of 𝑔.
8
3
𝑥
15
𝑔
7
5
𝑎•𝑏 = 𝑐•𝑑
(3) • (7) = (𝑥) • (5)
21 = 5𝑥
4.2 = 𝑥
𝑡2 = 𝑦(𝑦 + 𝑧)
152 = 8(8 + 𝑔)
225 = 64 + 8𝑔
161 = 8𝑔
20.125 = 𝑔
Ex.5: 2 Secants
• Find the length of 𝑥.
20
𝑤(𝑤 + 𝑥) = 𝑦(𝑦 + 𝑧)
14(14 + 20) = 16(16 + 𝑥)
(34)(14) = 256 + 16𝑥
476 = 256 + 16𝑥
220 = 16𝑥
13.75 = 𝑥
14
16
𝑥
Ex.6: A little bit of everything!
Solve for 𝒌 first.
𝑤(𝑤 + 𝑥) = 𝑦(𝑦 + 𝑧)
12
𝑘
175°
9(9 + 12) = 8(8 + 𝑘)
186 = 64 + 8𝑘
9
8
𝑎°
60°
𝑟
𝑘 = 15.6
Next solve for 𝒓
Lastly solve for 𝒎𝒂
𝑡2 = 𝑦(𝑦 + 𝑧)
𝑚1 = ½(𝑥 − 𝑦)
𝑟2 = 8(8 + 15.6)
𝑚𝑎 = ½(175 – 60)
𝑟2 = 189
𝑚𝑎 = 57.5°
𝑟 = 13.7
What have we learned??
• When dealing with angle measures formed by
intersecting secants or tangents you either add
or subtract the intercepted arcs depending on
where the lines intersect.
• There are 3 formulas to solve for segments
lengths inside of circles, it depends on which
segments you are dealing with: Secants,
Chords, or Tangents.
Challenge #1
D
6
C
30°
A
3
Q
5
2 1
4
100°
G
E
25°
𝐴𝐹 is a diameter,
𝑚𝐴𝐺 = 100°,
𝑚𝐶𝐸 = 30°
𝑚𝐸𝐹 = 25°
F
𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑎𝑙𝑙
𝑛𝑢𝑚𝑏𝑒𝑟𝑒𝑑 𝑎𝑛𝑔𝑙𝑒𝑠
Challenge #2
Find the value of the variable.
𝑧
8
6
7
16
𝑦
8
Challenge #3
Find the value of the variable.
20
14
6.5
3
𝑚
16
𝑥
7
D
6
C
30°
E
A
3
100°
Q
2 1
5
4
G
25°
F
m1  mFG  80
m2  m AG  100
55
1
 22.5
m3  (mCE  mEF ) 
2
2
80  155
1
 117.5
m4  (mGF  m ACE ) 
2
2
m5  180  117.5  62.5
1
1
m6  (m AG  mCE )  (100  30)  35
2
2
Find the value of the variable.
6
7
𝑧
8
𝑦
(6 + 8) 6 = (7 + 𝑦) 7
84 = 49 + 7𝑦
35 = 7𝑦
𝑦 = 5
16
8
𝑧2 = (16 + 8) 8
𝑧2 = 192
𝑧 = 13.9
Find the value of the variable to the nearest tenth.
6.5
20
3
14
16
𝑚
7
𝑥
(20 + 14) 14
476
220
𝑥
=
=
=
=
(𝑥 + 16) 16
16𝑥 + 256
16𝑥
13.8
6.5𝑚 = 3 ∙ 7
6.5𝑚 = 21
𝑚 = 3.2